The parameters that appear in the Lagrangian density (REF ) for the vortex are \(\lambda \) , \(e\) , \(n\) , \(v\) , \(\alpha \) , \(\Lambda \)  and \(\xi \) . The goal here is to determine how the vortex changes with the coupling \(\xi \)  and to observe what happens as we approach the critical couling \(\xi _c\) . How the vortex changes with the other parameters such as \(\Lambda \) , \(n\)  and \(v\)  has been studied elsewhere [1]}. We therefore run numerical simulations for different values of \(\xi \)  with the other parameters held fixed; we set \(\lambda =1\) , \(e=3\) , \(n=1\) , \(v=1\) , \(\alpha =1\) , and \(\Lambda =-1\) . We work in natural units where \(\hbar =c=1\) . Though our parameters and quantities such as the radius, mass and magnetic field are quoted as numbers, they should be thought of as having a unit attached to them (except for the winding number \(n\)  which is a pure number)All parameters and quantities can be expressed in terms of two independent units: a unit \(x\)  attached to the VEV \(v_{eff}\)  and a unit \(y\)  attached to the cosmological constant \(\Lambda _{eff}\) . The units \(x\)  and \(y\)  have dimensions of [L]\(^{-1/2}\)  and [L]\(^{-2}\)  respectively. It follows that the units attached to the parameter \(v\)  and \(\Lambda \)  are \(x\)  and \(y\)  respectively. From (REF ), the quantity \(-\Lambda _{eff}\,r^2\)  is dimensionless. Therefore the unit attached to the radius \(r\)  is \(y^{-1/2}\) . This can be expressed in terms of the AdS length \(\ell \) . We have that \(\Lambda _{eff} \times y =-1/\ell ^2\)  so that \(y^{-1/2}=(-\Lambda _{eff})^{1/2}\,\ell \)  where \(\Lambda _{eff}\)  here is a pure negative number. The equation for \(\epsilon (r)\)  in  implies that \(e\,v_{eff}/(-\Lambda _{eff})^{1/2}\)  must be dimensionless. The unit attached to the coupling \(e\)  is therefore \(x^{-1}\,y^{1/2}\,\)  and has dimensions [L]\(^{-1/2}\) . \(\lambda /e^2\)  is also dimensionless so that \(\lambda \)  is expressed in units of \(x^{-2}\,y\)  which has dimensions of [L]\(^{-1}\) . The mass is proportional to \(\alpha _{eff}= \alpha + v_{eff}^2 \,\xi \)  and therefore has units of \(x^2\)  i.e. the unit of the VEV squared which is the same unit as in fixed Minkowski spacetime [2]}. This has dimensions of [L]\(^{-1}\)  which is the correct dimension for the mass in natural units. The magnetic field, given by \(B_m= \frac{\sqrt{A}\, a^{\prime }}{e \,r}\)  has units of \(x\,y^{1/2}\)  which has dimensions of [L]\(^{-3/2}\) . Note that \(A(r)\)  and \(a(r)\)  are dimensionless.. As we have seen, a negative \(\Lambda \)  automatically ensures that the asymptotic cosmological constant \(\Lambda _{eff}\)  will be negative. Our solutions in this section will therefore correspond to an AdS\(_3\)  background. Note that though \(v\)  and \(\Lambda \)  are held fixed, the VEV \(v_{eff}\)  and the cosmological constant \(\Lambda _{eff}\)  will change with \(\xi \) .
